Category Archives: Statistical mechanics

This is the second part of a study on the statistical nature of sound.

The incident velocity of the collision

Let's consider a particle moving along the x-axis towards the right, in an isolated system with zero average particle velocity. A single particle, moving towards the right, among several particles moving according to the statistical distribution of ideal gases shown in the previous post. Now let's attach the reference point to the particle. This will be equivalent to a situation where the particle stays still, and all the other particles follow a shifted velocity distribution. The probability distribution will be shifted by the velocity of the single particle, but in in the opposite direction.

Note that as we are now investigating things from the perspective of the particle, all collisions will be identical, no matter what the direction of the incoming particle is. Only the velocity of the incoming particle and the angle of incidence matters. This is illustrated in the following image; we can rotate the possible collisions coming from a certain direction any way we like, the situation stays symmetric as the particle doesn't have any velocity from the perspective of the particle.

Collisions only depend on velocity and angle of incidence of other particles if we consider the situation from the perspective of a single particle

Now we have two things to sort out: how fast will the incoming particle move, and what angle will it arrive from?

Almost all the information we require to solve this is shown in the shifted probability distribution. As an intuitive example: the further we shift the probability distribution to the left, the larger portion of the distribution will be to the left of the particle. In practice this means that when a single particle moves fast enough to the right, almost all particles it encounters will move to the left (relative to the moving particle).

The faster some other particle moves, the more probable it is that it will hit the static particle we're observing within a given time. This relationship is very nearly linear, which can be based on the following: the area a single particle, somewhere in the box, will cover in a given time is almost directly proportional to its speed. Thus the chance that the area will overlap the area required for a collision is also directly proportional to its speed (except at very small velocities, which we can safely ignore here, as the average distance between collisions is significantly larger than the area of one particle, these small velocity collisions happen very rarely and almost no kinetic energy is transferred when they happen). This probabilistic relationship is shown in the figure below. The figure shows a random particle and the probability that it will hit another particle, purely on the basis of its velocity. Note that this figure will approach infinity if there are an infinite number of velocities (the probability will continue rising forever with increasing velocity), so the statistical distribution will always need to be confined into some specific range of velocities.

The relative probability of another randomly distributed particle hitting our particle, as a function of velocity

Note also that the statistical distribution shown to the left is purely based on velocity. It doesn't take into account the distribution of velocities (the chance that a random particle has a certain velocity, based on the Maxwell speed distribution). The distribution of the velocities is taken into account in the shifted probability distribution.

To get the probability distribution for how likely it is that a random particle with some velocity will hit our particle, we do the following: we multiply the shifted particle velocity distribution with the probability distribution based purely on velocity (and normalize this distribution so that the statistics are valid, but never mind this now). This can be seen, in the following figure, to show what intuitively can be expected; if the particle is moving to the right, it is much more likely that a collision will happen with a particle which is moving to the left relative to the particle. Note that this statistical distribution will look different depending on the velocity of the particle we are investigating!

Probability of a random particle colliding with a static particle if a certain average particle velocity exists, given the velocity of the random particle

Considering the phenomena of sound propagation at the very basic level, I think, is the best way for me to understand the very nature of sound. I'm planning on investigating the nature of sound in gases at standard temperature and pressure, yet resorting to as little math as possible (to keep things as clear as I can). I think surprisingly many attributes of sound, which otherwise might be difficult to grasp, can be explained from a statistical point of view (this area of physics is called statistical mechanics). And then there's always the saying "If you can't explain it simply, you don't understand it well enough". So I shall give it a try! Simplicity can also be relative, something to keep in mind 🙂

An ideal gas

I'll start from the assumption that sound propagates in an ideal gas. I've found that the concept of an ideal gas is best understood through the kinetic theory of gases.

The movement of molecules in an ideal gas, according to the kinetic theory of gases (source: Wikipedia)

By viewing the great picture provided by Wikipedia, one can see that an ideal gas consists of particles (molecules) which are constantly in motion. By isolating the system (like in the picture) and assuming that no energy is lost in any of the collisions, the particles will continue colliding forever. This seems like a starting point which is relatively easy to understand.

A small portion of air, like in the box, contains a certain amount of kinetic energy, distributed throughout the particles in the form of different velocities. The velocities of the particles in an isolated system will follow a statistical distribution called the Maxwell speed distribution. I will not investigate the derivation of this distribution any further, and just assume it holds true.

Probability distribution of the velocity of a single particle

We can illustrate the statistical distribution further by doing a plot of the most probable velocity a random particle will have in the system. We place the particle in the middle, and consider the color surrounding it as describing the most probable place to draw a velocity vector, with red corresponding to the most probable choice and purple the least probable choice.

There are a few familiar ways to measure the amount of kinetic energy stored in the isolated particle system:

pressure, which describes the average force per unit area, caused by the particles colliding with the walls of the container.

temperature, which describes the average kinetic energy of the particles.

Probability distribution of the velocity of a single particle with an average particle velocity to the left

There's also another really important concept here which relates to acoustics; the average velocity of the particles. In the case of the isolated system above, the average particle velocity will be 0, as the velocities are equally distributed in all directions. But if the box was moving relative to some fixed point, the average particle velocity would differ from 0, and instead be equal to the velocity of the box. This is illustrated in the figure to the right, where the added average velocity simply shifts the probability distribution to some direction by the amount of the average particle velocity. The average velocity is relative; it's fully dependent on the point of reference.